1 Algebraic preliminaries
We gather in this section some standard facts about Clifford algebras and their rep-
resentations. For details and proofs we refer to [LM] or [Kal].
§1.1 Clifford algebras Let Q be a quadratic form on V. The Clifford algebra
generated by V and Q, denoted by C(V, Q) is the associative unital algebra generated
by V with the relations
u - v -f v u = —Q(u, v) - 1 V u, v 6 V
If ei, •, en (71 = dim V) is a basis of V in which Q is diagonal then C(V, Q) can be
alternatively characterized as the associative unital algebra generated by ei,---,e
modulo the relations
e{ 6j + e3;• et- = -22(et-,ej) V ij. (1.1)
For any nonnegative integers p, / such that p -f ? 0 we denote by Rp'9 the space
W © M9 endowed with the quadratic form
Q(x®y) = \x\2-\y\2 xeW , yeRq
where | | denotes the standard euclidian metric. Then Cp,q denotes the Clifford
algebra generated by Rp'9. When p = q = 0 we set C°'° = R.
Let e-i, ep; cj, tq denote the standard basis of W,g. Cp,g decomposes as a
Z2-graded algebra ("superalgebra")
r,q & C™®CpJq (1.2)
where C±q are the vector subspaces generated by the even/odd degree monomials in
the basis elements {et; tj).
Cp,q can be naturally equipped with a scalar product making {ej ej] an or-
thonormal basis. Denote the corresponding norm by || ||. (Cp'9, || ||) is a Z2-graded
real C*-algebra with the anti-involution "*" uniquely defined by its action on the
e* = -ei , e* = t3 V ij.
It will be useful to introduce some "super" notions.
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